lsatcvat - Mathbox for Norm Megill

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Theorem lsatcvat 39852

Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32747 analog.) (Contributed by NM, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
lsatcvat.o	⊢ 0 = (0_g‘𝑊)
lsatcvat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsatcvat.p	⊢ ⊕ = (LSSum‘𝑊)
lsatcvat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatcvat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat.n	⊢ (𝜑 → 𝑈 ≠ { 0 })
lsatcvat.l	⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat	⊢ (𝜑 → 𝑈 ∈ 𝐴)

**Proof of Theorem lsatcvat**
Step	Hyp	Ref	Expression
1		lsatcvat.o	. . 3 ⊢ 0 = (0_g‘𝑊)
2		lsatcvat.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lsatcvat.p	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
4		lsatcvat.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lsatcvat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	5	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec)
7		lsatcvat.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
8	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
9		lsatcvat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
11		lsatcvat.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
13		lsatcvat.n	. . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 })
14	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 })
15		lsatcvat.l	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
16	15	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
17		simpr 489	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈)
18	1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17	lsatcvatlem 39851	. 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
19	5	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec)
20	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
21	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
22	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
23	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 })
24		lveclmod 21236	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
25	5, 24	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
26		lmodabl 21039	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
27	25, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel)
28	2	lsssssubg 21088	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
29	25, 28	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	2, 4, 25, 9	lsatlssel 39799	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
31	29, 30	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
32	2, 4, 25, 11	lsatlssel 39799	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
33	29, 32	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
34	3	lsmcom 19932	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
35	27, 31, 33, 34	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
36	35	psseq2d 4050	. . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄)))
37	15, 36	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
38	37	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
39		simpr 489	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈)
40	1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39	lsatcvatlem 39851	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
41	29, 7	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
42	3	lsmlub 19738	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
43	31, 33, 41, 42	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
44		ssnpss 4061	. . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))
45	43, 44	biimtrdi 256	. . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)))
46	45	con2d 135	. . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈)))
47		ianor 997	. . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
48	46, 47	imbitrdi 254	. . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)))
49	15, 48	mpd 16	. 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
50	18, 40, 49	mpjaodan 973	1 ⊢ (𝜑 → 𝑈 ∈ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ⊆ wss 3905 ⊊ wpss 3906 {csn 4589 ‘cfv 6536 (class class class)co 7410 0_gc0g 17496 SubGrpcsubg 19190 LSSumclsm 19708 Abelcabl 19855 LModclmod 20990 LSubSpclss 21061 LVecclvec 21232 LSAtomsclsa 39776

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-0g 17498 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-cntz 19391 df-oppg 19420 df-lsm 19710 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-drng 20838 df-lmod 20992 df-lss 21062 df-lsp 21102 df-lvec 21233 df-lsatoms 39778 df-lcv 39821

This theorem is used by: lsatcvat2 39853

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